Kernel density estimation on random fields

On smoothed probability density estimation for stationary processes

Recursive estimation of the univariate probability density function f(x) for stationary processes \{X_{j}\} is considered. Quadratic-mean convergence and asymptotic normality for density estimators f_{n}(x) are established for strong mixing and for asymptotically uncorrelated processes \{X_{j}\} . Recent results for nonrecursive density estimators are extended to the recursive case.

Recursive probability density estimation for weakly dependent stationary processes

Strong convergence of sums of α-mixing random variables with applications to density estimation

A local linear kernel estimator of the regression function x → g(x):= E[Y i |X i = x], x ∈ R d , of a stationary (d + 1)-dimensional spatial process {(Y i , X i ), i ∈ Z N } observed over a rectangular domain of the form in:= {i = (i 1 ,..,i N ) ∈ Z N |1 ≤ i k ≤ n k , k = 1,...,N}, n = (n 1 ,...,n N ) ∈ Z N , is proposed and investigated. Under mild regularity assumptions, asymptotic normality of the estimators of g(x) and its derivatives is established. Appropriate choices of the bandwidths are proposed. The spatial process is assumed to satisfy some very general mixing conditions, generalizing classical time-series strong mixing concepts. The size of the rectangular domain in is allowed to tend to infinity at different rates depending on the direction in Z N .

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Local linear spatial regression

For broad classes of deterministic and random sampling schemes \{t_{k}\} we establish the consistency and asymptotic expressions for the bias and covariance of discrete-time estimates f_{n}(x) for the marginal probability density function f(x) of continuous-time processes X(t) . The effect of the sampling scheme and the sampling rate on the performance of the estimates is studied. The results are established for continuous-time processes X(t) satisfying various asymptotic independence-uncorrelatedness conditions.

E. Masry

Papers

Probability density estimation from sampled data